MHB Proving Root Space Invariancy of Linear Transformation

[Sudharaka]

Hi everyone, :)

Here's a question that I don't quite understand.

Given \(f:\, V\rightarrow V\) and a root space \(V(\lambda)\) for \(f\), prove that \(V(\lambda)\) is invariant for \(g:,V\rightarrow V\) such that \(g\) commutes with f.

What I don't understand here is what is meant by root space in the context of a linear transformation. Can somebody please explain this to me or direct me to a link where it's explained?

 

[Opalg]

It looks as though a root space is what I would call an eigenspace, in other words the subspace of all eigenvectors corresponding to a given eigenvalue $\lambda$. (Strictly speaking, an eigenvector has to be nonzero, so the eigenspace is the set of all eigenvectors corresponding to $\lambda$, together with the zero vector.)

 

[Sudharaka]

It looks as though a root space is what I would call an eigenspace, in other words the subspace of all eigenvectors corresponding to a given eigenvalue $\lambda$. (Strictly speaking, an eigenvector has to be nonzero, so the eigenspace is the set of all eigenvectors corresponding to $\lambda$, together with the zero vector.)

Thanks very much for your valuable reply. :) There was some doubt on my mind as to what this root space is all about. It seems to me that there is some ambiguity about this depending on different authors. For example I was reading the following and it has a slightly different definition about the root space.

Matrix And Linear Algebra 2Nd Ed. - Datta - Google Books

However I don't know what my prof. had in his mind when writing down this question. So to make matters simple I shall take the eigenspace as the rootspace. Thanks again, and have a nice day. :)

 

[Opalg]

I was reading the following and it has a slightly different definition about the root space.

Matrix And Linear Algebra 2Nd Ed. - Datta - Google Books

Probably the definition in the book by Datta is the standard one. My previous suggestion was only a guess.